
// Make newform 170.2.o.b in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_170_o();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_170_2_o_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_170_o();"
function MakeCharacter_170_o()
    N := 170;
    order := 16;
    char_gens := [137, 71];
    v := [12, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_170_o_Hecke(Kf)
    return MakeCharacter_170_o();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_170_2_o_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_170_2_o_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_170_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![524288, -6815744, -13828096, 178978816, 1316325376, 3709325312, 7750865920, 10429304576, 13965074692, 10616388960, 7954431568, 126285856, 2699236504, 420903552, 1480893040, -1118637168, 2425345800, -1312500256, 190182008, 223397360, -315770980, 179730448, -52963904, -37440936, 36096213, -4096072, -1284088, 418920, 215228, -46472, -9224, 3864, 576, -376, 136, 72, -48, -8, 0, 0, 1]>],Snew);
    return Vf;
end function;